I’m continuing to work through the fantastic Apostolic Fathers: An Introduction edited by Wilhelm Pratscher, and one of the truly fantastic parts of this book is that every chapter has a section on intertextuality. Besides being a generally fascinating subject, it is important for how we conceive of Christianity in the earliest of the early church. In the chapter on 1 Clement written by Andreas Lindemann, the relationship between 1 Clement and Hebrews is briefly explored:

The special similarity between 1Clem [sic] and Hebrews was recognized even in the early church. Eusebius writes (HEIII 38.1-3) that the fact that the author of 1Clem cites Hebrews means that Hebrews cannot be a “young” text; he also refers to the similarity of style and thought. Origen concluded from [i.e. according to] Eus VI 25.14 that people saw Clement of Rome (or even Luke) as the author of Hebrews. (pg. 59)

The section of Eusebius’ Ecclesiastical History Lindemann refers to is specifically dealing with what Origen has argued about the nature and authorship of the scriptures. It contains some interesting features like the claim of Matthean priority and the denial of the authenticity of 2 Peter. The section referred to above says:

11. In addition he makes the following statements in regard to the Epistle to the Hebrews in his Homilies upon it: “That the verbal style of the epistle entitled ‘To the Hebrews,’ is not rude like the language of the apostle, who acknowledged himself ‘rude in speech’ but that its diction is purer Greek, any one who has the power to discern differences of phraseology will acknowledge.
12. Moreover, that the thoughts of the epistle are admirable, and not inferior to the acknowledged apostolic writings, any one who carefully examines the apostolic text will admit.”
13. Farther on he adds: “If I gave my opinion, I should say that the thoughts are those of the apostle, but the diction and phraseology are those of some one who remembered the apostolic teachings, and wrote down at his leisure what had been said by his teacher. Therefore if any church holds that this epistle is by Paul, let it be commended for this. For not without reason have the ancients handed it down as Paul’s.
14. But who wrote the epistle, in truth, God knows. The statement of some who have gone before us is that Clement, bishop of the Romans, wrote the epistle, and of others that Luke, the author of the Gospel and the Acts, wrote it.” But let this suffice on these matters. (Schaff, Eus VI 25.11-14)

Lindemann continues:

To be sure, neither agreement by citation nor other references to the text allow for the assumption of a direct literary relationship. But the similarity of 1Clem 36.2-5 to Hebrews 1:3-5, 7, 13 is so great that the use of Hebrews by 1Clem must still be considered possible.

Placing the texts side by side the similarity is obvious.

1 Clement 36.2-5

Hebrews 1:3-5, 7, 13

2By Him we look up to the heights of heaven. By Him we behold, as in a glass, His immaculate and most excellent visage. By Him are the eyes of our hearts opened. By Him our foolish and darkened understanding blossoms up anew towards His marvelous light. By Him the Lord has willed that we should taste of immortal knowledge, who, being the brightness of His majesty, is by so much greater than the angels, as He hath by inheritance obtained a more excellent name than they.3 For it is thus written, Who maketh His angels spirits, and His ministers a flame of fire.

4 But concerning His Son the Lord spoke thus: Thou art my Son, today have I begotten Thee. Ask of Me, and I will give Thee the heathen for Thine inheritance, and the uttermost parts of the earth for Thy possession.

5 And again He saith to Him, Sit Thou at My right hand, until I make Thine enemies Thy footstool.

3He is the reflection of God’s glory and the exact imprint of God’s very being, and he sustains all things by his powerful word. When he had made purification for sins, he sat down at the right hand of the Majesty on high,4 having become as much superior to angels as the name he has inherited is more excellent than theirs. 5 For to which of the angels did God ever say, “You are my Son; today I have begotten you”? Or again, “I will be his Father, and he will be my Son”?

7 Of the angels he says, “He makes his angels winds, and his servants flames of fire.”

13 But to which of the angels has he ever said, “Sit at my right hand until I make your enemies a footstool for your feet”?

Ante-Nicene Fathers

NRSV

Lindemann notes some differences that detract from the idea of a close relationship between texts. Specifically, both 1 Clement and Hebrews use the language of “high priest” to describe Jesus, but they do so in divergent ways. Lindemann suggests that it is possible that the author of 1 Clement knew Hebrews 1, but not the rest of the letter. All in all, it is a rather complicated issue. There seems to be a clear relationship, but the extent of the relationship is anything but clear.